The dominant-pair conjecture for singular random sign matrices

Let $M_n$ be an $n\times n$ random matrix whose entries are independent and uniformly distributed in $\{+1,-1\}$. A matrix is singular when its determinant vanishes. It is known that matrices with two identical or opposite rows or columns are singular, giving a lower bound of order $n^2 2^{1-n}$ for the singularity probability.

Dominant-pair conjecture.

$$\mathbf{P}(\det M_n=0)=(1-o(1))n^2 2^{1-n}.$$

The conjecture asserts that pairs of identical or opposite rows or columns provide the dominant source of singularity for random $\pm1$ matrices. Komlós had already shown that the singularity probability tends to zero, but determining its precise asymptotic was an open problem in the source paper.